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A note on theta pairs for maximal subgroups. (English) Zbl 0919.20013

Let \(M\) be a maximal subgroup of a finite group \(G\). A pair of subgroups \((C,D)\) of \(G\) is called a \(\theta\)-pair of \(M\) if it satisfies the following conditions: (a) \(D\leq C\) and \(D\vartriangleleft G\); (b) \(\langle M,C\rangle=G\) and \(D\leq M\); (c) \(C/D\) has no proper normal subgroup of \(G/D\). A \(\theta\)-pair \((C,D)\) of \(M\) is said to be maximal if \(M\) has no \(\theta\)-pair \((C',D')\) such that \(C<C'\). Several results are obtained on maximal \(\theta\)-pairs which imply \(G\) to be solvable or supersolvable.

MSC:

20D25 Special subgroups (Frattini, Fitting, etc.)
20E28 Maximal subgroups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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